226 CHEMICAL ENGINEERING For horizontal expansion chambers of regular form, such as truncated pyramids or cones, the path of a particle may be calculated theoretically from the following formula:1 where A is the cross-sectional area of the small end of the chamber: A' the cross-sectional area of the large end; L the length of the chamber; V the terminal velocity of the current in feet per second equal to vAJA', where v is the entry velocity of the current; g the constant of acceleration and c a factor to apply to the fall velocity in vacuo. The origin, of the curve is taken through the particle at the point where it first comes under the influence of the blast. This mode of grading is used on feathers. It is believed by the writer to be used in the dry treatment of mica. The only plant of this kind of any importance is said to use it but details of the means employed cannot be obtained. Ground mica is used in the manufacture of wall papers, lubricants, fancy paints, molded blocks for electrical insulation. About eight sizes of ground mica are made ranging from 10 mesh to bran mica, Grading by Blowing Over Plane Surfaces.—This operation is usually a separation of coarse from fine but it might be made susceptible to obtaining more than two sizes by means for scraping up the gradings made as they are blown to their different positions. Friction is the controlling factor with this ' sort of a device. Since the weight of the particle increases as the cube of the diameter, but the surface exposed to the blast only as the square the finer particles are carried farther along the surface than the coarser ones. Particles of superior specific gravity are carried a less distance than ones of the same size but of less specific gravity. The operation as to a mere separation is analogous to that performed by a fanning machine for separating chaff from wheat or cereals. The final separation of black sand from gold is done by Mexicans by blowing the dry sand away from the gold contained in a batea or dry gold pan. Grading by Projection.—-Borne attempt has been made to grade material by taking advantage of the greater or less distance to which particles are projected from the end of an endless belt. In using this principle advantage is taken of the fact that owing to the opposition of the air the largest particles In the last two formulae on p. 22/> <{ {» the density of the air at the pressure -p. c is the average velocity of agitation of the air molecule. i\ is the coefficient of viscosity of the air at chosen temperature. For those who wish to pursue the subject of fall velocity further the following references may be of value. The case in which the velocity is only slightly greater than the critical value, i.e.., that value at which the firstpower law of resistance begins to break down, was treated by AHNOLD in the Philosophical Magazine for about 1911 and 1012. Then* is also .some experimental work on fall velocities for greater speeds in the same publication for the years IOH, li)lf> and I'JKi. As to the time required to reach the equilibrium speed where the first power law of velocity ultimately holds, this is exceedingly simple. A particle which falls in air at a rate of say, 1 cm. a second will reach its steady rate of fall in much less than one-thousandth of a second. That problem has been worked out by Rayleitfh. and can be found discussed in an article of JOHN ZKLKNY'H in the Phijsicdl Itcvific for I'.HO under the heading "The Terminal Velocity of Small Spheres in Air." 1 The horizontal velocity at any point, distant, x from the origin, in the expansion ehamlj obtained from the proportion V: vx: : x: L or v equals . Now by the; theoretical mechanics equals jr. dt LV v or — and consequently xdv equals LVdt. On integrating both sides of this equation there is obtained & /2/A ^ the equation K^"2 equals LVt or t equals -----. In time t the particle has settled the distance el - I • 2L V \ g / On equating the two equations equal to the time t arid transposing the expression above is obtained. It